Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $t = \dfrac{p + 3}{p^2 + 2p - 35} \times \dfrac{-7p + 35}{p + 3} $
Answer: First factor the quadratic. $t = \dfrac{p + 3}{(p - 5)(p + 7)} \times \dfrac{-7p + 35}{p + 3} $ Then factor out any other terms. $t = \dfrac{p + 3}{(p - 5)(p + 7)} \times \dfrac{-7(p - 5)}{p + 3} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (p + 3) \times -7(p - 5) } { (p - 5)(p + 7) \times (p + 3) } $ $t = \dfrac{ -7(p + 3)(p - 5)}{ (p - 5)(p + 7)(p + 3)} $ Notice that $(p + 3)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ -7(p + 3)\cancel{(p - 5)}}{ \cancel{(p - 5)}(p + 7)(p + 3)} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $t = \dfrac{ -7\cancel{(p + 3)}\cancel{(p - 5)}}{ \cancel{(p - 5)}(p + 7)\cancel{(p + 3)}} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $t = \dfrac{-7}{p + 7} ; \space p \neq 5 ; \space p \neq -3 $